By Balakrishnan A.V.

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**Example text**

Using this and the nonnegativity of the Taylor coeﬃcients of the functions T1 (z)α and 1/(1 − z 2 ) we get 1 + cz 1−z α−1 1 1 = ϕ1 (z)α 2 (1 − z) 1 − z2 1+z 1−z α 1 . 18. Now, we prove a subordination theorem which we need for the applications. We are concerned with the class of angular domains Πα = aHα + b (a = 0) with opening angle απ, 1 ≤ α ≤ 2, which means that there exists a linear transformation T (z) = az + b such that Πα = T (Hα ), where Hα = z | | arg z| < απ . 2 Clearly, the assertion of the following theorem is a generalization of the Carath´eodory inequality for Taylor coeﬃcients of holomorphic functions with positive real part (see [51], [52], compare also [42], p.

0 e−int dμ(t). (1 − ζe−it )n+1 Consequently, by simple estimates and the Poisson formula |f (n) (z)| ≤ 2π 2n! (1 − |z|)n−1 (1 − |z|2 ) 0 2n! Ref (z) 1 − |z|2 dμ(t) = . |1 − ze−it |2 (1 − |z|)n−1 (1 − |z|2 ) Since 1 − |z| = |1 − ze−it | is true only for e−it = e−it0 = z/|z|, it is easily seen that equality can occur only for a piece-wise constant function μ(t) such that μ([0, 2π]) = {0, 1}, and, consequently, the corresponding function f0 has the form f0 (ζ) = 1 + ζe−it0 , 1 − ζe−it0 eit0 = z/|z|, and f0 maps Δ onto H1 .

1 4 for We will need Landau’s theorem on holomorphic functions that omit two ﬁxed values. More precisely, the Landau theorem concerns functions ∞ an z n f (z) = n=0 holomorphic in the unit disc Δ and omitting the values 0 and 1 in Δ. By our notation, f ∈ A(Δ, C \ {0, 1}). 3768796 . . 4π 2 is used. J. A. Hempel and J. A. Jenkins established an explicit sharp bound in the Landau theorem. 34 Chapter 3. 10 (Landau; see J. A. Hempel [80] and J. A. Jenkins [87] for proofs). If the function f is holomorphic and omits 0 and 1 in Δ, then |a1 | ≤ 2|a0 | |log |a0 || + Γ(1/4)4 4π 2 .